Material that uses Texture Coordinates won't "stick" to meshes in a physics simulation
Duarte Farrajota Ramos really deserves credit for this, but the solution is as follows:

In the UV Editing workspace, select mesh in Object mode.
Switch to Edit mode and select all vertices in the mesh (A).
Press U to unwrap mesh. (If it has already been incorrectly unwrapped, delete the existing UVMap from the Object Data Properties, one of the tabs in the Properties Editor on the right).
Choose an unwrapping method from the menu. In this case, Cube Projection worked best.
Change Texture Coordinate node output from Object to UV.